To see what the symmetries (REF , REF ) imply, let us examine the solutions of (REF ). For our purposes, it is sufficent to solve (REF ) near the bang and the dS boundary, where the gravitational backreaction of \(\varphi \)  can be neglected. More generally, the solution \(\tilde{\varphi }\)  is an arbitrary linear combination of separable solutions \(\tilde{\varphi }_{k}(\eta )\psi _{k}({\bf x})\) , where \(\psi _{k}({\bf x})\)  is an eigenfunction of
\(\gamma ^{ij}\tilde{\nabla }_{i}\tilde{\nabla }_{j}\) ; and, if we neglect the gravitational backreaction of \(\varphi \) , so that \(a(\eta )\)  is given by (REF ), then (REF ) becomes the Lamé equation for \(\tilde{\varphi }_{k}(\eta )\) , which can be solved in terms of Lamé functions [1]}.
